Difference between revisions of "2014 AMC 12B Problems/Problem 15"
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==Problem== | ==Problem== | ||
− | When <math>p = \sum\limits_{k=1}^{6} k \ln{k}</math>, the number <math>e^p</math> is an integer. What is the largest power of 2 that is a factor of <math>e^p</math>? | + | When <math>p = \sum\limits_{k=1}^{6} k \text{ ln }{k}</math>, the number <math>e^p</math> is an integer. What is the largest power of 2 that is a factor of <math>e^p</math>? |
− | <math> \textbf{(A)}\ 2^{12}\qquad\textbf{(B)}\ 2^{14}\qquad\textbf{(C)}\ 2^{16}\qquad\textbf{(D) | + | <math> \textbf{(A)}\ 2^{12}\qquad\textbf{(B)}\ 2^{14}\qquad\textbf{(C)}\ 2^{16}\qquad\textbf{(D)}\ 2^{18}\qquad\textbf{(E)}\ 2^{20} </math> |
==Solution== | ==Solution== | ||
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This product has <math>2</math> powers of <math>2</math> in the <math>2^2</math> factor, <math>4*2=8</math> powers of <math>2</math> in the <math>4^4</math> factor, and <math>6</math> powers of <math>2</math> in the <math>6^6</math> factor. | This product has <math>2</math> powers of <math>2</math> in the <math>2^2</math> factor, <math>4*2=8</math> powers of <math>2</math> in the <math>4^4</math> factor, and <math>6</math> powers of <math>2</math> in the <math>6^6</math> factor. | ||
This adds up to <math>2+8+6=16</math> powers of two which divide into our quantity, so our answer is <math>\boxed{\textbf{(C)}\ 2^{16}}</math> | This adds up to <math>2+8+6=16</math> powers of two which divide into our quantity, so our answer is <math>\boxed{\textbf{(C)}\ 2^{16}}</math> | ||
+ | |||
+ | ==Video Solution== | ||
+ | |||
+ | For those wanting a video: https://www.youtube.com/watch?v=iq2X86GFVBo | ||
+ | == See also == | ||
+ | {{AMC12 box|year=2014|ab=B|num-b=14|num-a=16}} | ||
+ | {{MAA Notice}} |
Latest revision as of 13:26, 26 November 2020
Contents
Problem
When , the number is an integer. What is the largest power of 2 that is a factor of ?
Solution
Let's write out the sum. Our sum is equal to Raising to the power of this quantity eliminates the natural logarithm, which leaves us with This product has powers of in the factor, powers of in the factor, and powers of in the factor. This adds up to powers of two which divide into our quantity, so our answer is
Video Solution
For those wanting a video: https://www.youtube.com/watch?v=iq2X86GFVBo
See also
2014 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 14 |
Followed by Problem 16 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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